In the figure below, equilateral
triangles ABD and BCE are drawn on the sides of a triangle ABC.
F, G, and H are the midpoints of AD, CE, and AC respectively. HL
and FG are perpendicular. Lines FH, CD, HL, and FG are cut
by line AE at J, O, P, and N respectively. Lines GH, HL, and FG
are cut by line CD at K, Q, and M respectively. Prove the
following: